A035485 Card on top of deck at n-th stage of R. K. Guy's shuffling problem.

1, 2, 3, 1, 6, 5, 9, 1, 4, 2, 16, 10, 12, 14, 23, 16, 18, 20, 17, 27, 30, 33, 38, 10, 14, 37, 32, 6, 11, 19, 53, 37, 25, 21, 12, 34, 38, 8, 50, 48, 46, 14, 18, 23, 47, 53, 84, 52, 31, 49, 1, 51, 91, 61, 42, 79, 4, 29, 6, 49, 26, 23, 115, 4, 70, 93, 109, 11, 16, 19, 49, 18, 124, 97, 70, 10, 134, 111, 7, 38, 14, 79, 11, 129

Offset: 0

Author

N. J. A. Sloane, Clark Kimberling

Keywords

• nonn
• easy
• nice

Comments

At n-th step, pick up top n cards and interlace them with the next n.

Here is the deck after steps 0,1,2,3,4,5:

1,2,3,4,5,6,7,...

2,1,3,4,5,6,7,...

3,2,4,1,5,6,7,...

1,3,5,2,6,4,7,8,9,...

6,1,4,3,7,5,8,2,9,10,...

It is conjectured that eventually every number appears on top of the deck.

See A035491 for (the relevant part of) the deck after the n-th step. - M. F. Hasler, Aug 13 2022

References

• D. Gale, Mathematical Entertainments: "Careful Card-Shuffling and Cutting Can Create Chaos," The Mathematical Intelligencer, vol. 14, no. 1, 1992, pages 54-56.
• D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998.

Links

• Lars Blomberg, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Perfect Shuffle.

Crossrefs

See A035491 for the array, also A035490, A035492.

Programs

PARI

A035485(n)=A035491_row(n+!n)[1]-!n \\ M. F. Hasler, Aug 13 2022

Python

def aupton(terms):
  alst, deck = [1], list(range(1, 2*terms+1))
  for n in range(1, terms+1):
    first, next = deck[:n], deck[n:2*n]
    deck[0:2*n:2] = next
    deck[1:2*n:2] = first
    alst.append(deck[0])
  return alst
print(aupton(83)) # Michael S. Branicky, Feb 01 2021

Formula

a(n) = A035491(n,1), i.e., the first element of the n-th row of that table, for all n > 0. - M. F. Hasler, Aug 13 2022

Extensions

More terms from Jud McCranie